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Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.

Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory.

Treatment

Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continuous, any mix of these two and more.

Classical definition: Initially the probability of an event to occur was defined as number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space.

For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by , since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.

Modern definition:

The modern definition starts with a set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by . It is then assumed that for each element , an intrinsic "probability" value is attached, which satisfies the following properties:

That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and the sum of f(x) over all values x in the sample space Ω is equal to 1. An event is defined as any subset of the sample space . The probability of the event is defined as

So, the probability of the entire sample space is 1, and the probability of the null event is 0.

The function mapping a point in the sample space to the "probability" value is called a probability mass function abbreviated as pmf. The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence.

Continuous probability distributions

Continuous probability theory deals with events that occur in a continuous sample space.

Classical definition: The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox.

Modern definition: If the outcome space of a random variable X is the set of real numbers () or a subset thereof, then a function called the cumulative distribution function (or cdf) exists, defined by . That is, F(x) returns the probability that X will be less than or equal to x.

In case the probability density function exists, this can be written as

Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables (including discrete random variables) that take values in

These concepts can be generalized for multidimensional cases on and other continuous sample spaces.

Measure-theoretic probability theory

The raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.

An example of such distributions could be a mix of discrete and continuous distributions, for example, a random variable which is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a pdf of , where δ[x] is the Dirac delta function.

Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space:

Given any set , (also called sample space) and a σ-algebra on it, a measure defined on is called a probability measure if

If is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on for any cdf, and vice versa. The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables, and pdf for continuous variables, making the measure-theoretic approach free of fallacies.

The probability of a set in the σ-algebra is defined as

where the integration is with respect to the measure induced by

Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside , as in the theory of stochastic processes. For example to study Brownian motion, probability is defined on a space of functions.

Convergence of random variables

In probability theory, there are several notions of convergence for random variables. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions.

Weak convergence: A sequence of random variables converges weakly to the random variable if their respective cumulative distribution functions converge to the cumulative distribution function of , wherever is continuous. Weak convergence is also called convergence in distribution.

Most common short hand notation:

Convergence in probability: The sequence of random variables is said to converge towards the random variable in probability if for every ε > 0.

Most common short hand notation:

Strong convergence: The sequence of random variables is said to converge towards the random variable strongly if . Strong convergence is also known as almost sure convergence.

Most common short hand notation:

As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies weak convergence. The reverse statements are not always true.

Law of large numbers

Common intuition suggests that if a fair coin is tossed many times, then roughly half of the time it will turn up heads, and the other half it will turn up tails. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of heads to the number of tails will approach unity. Modern probability provides a formal version of this intuitive idea, known as the law of large numbers. This law is remarkable because it is nowhere assumed in the foundations of probability theory, but instead emerges out of these foundations as a theorem. Since it links theoretically-derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory.[1]

The law of large numbers (LLN) states that the sample average of (independent and identically distributed random variables with finite expectation μ) converges towards the theoretical expectation μ.

It follows from LLN that if an event of probability p is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards p.

Putting this in terms of random variables and LLN we have are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1-p. E(Yi) = p for all i and it follows from LLN that converges to palmost surely.

Central limit theorem

"The central limit theorem (CLT) is one of the great results of mathematics." (Chapter 18 in [4].) It explains the ubiquitous occurrence of the normal distribution in nature.

The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distributionirrespective of the distribution followed by the original random variables. Formally, let be independent random variables with mean and variance Then the sequence of random variables

Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or happen over time in an apparently random fashion.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.

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History

The mathematical theory of probability has its roots in attempts to analyse games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points"). Although an individual coin toss or the roll of a die is random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.